Question: Determine how many solutions exist for the system of equations. ${x-y = 9}$ ${-10x+2y = 12}$
Answer: Convert both equations to slope-intercept form: ${x-y = 9}$ $x{-x} - y = 9{-x}$ $-y = 9-x$ $y = -9+x$ ${y = x-9}$ ${-10x+2y = 12}$ $-10x{+10x} + 2y = 12{+10x}$ $2y = 12+10x$ $y = 6+5x$ ${y = 5x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-9}$ ${y = 5x+6}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.